What is a group in group theory?

A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for every element.

What are the main properties of a group?

The four main properties are: Closure: The result of the operation on any two elements of the group is also in the group. Associativity: The group operation is associative. Identity: There exists an element that does not change other elements when used in the operation. Invertibility: Every element has an inverse that, when combined with the element, yields the identity.

What is the identity element in a group?

The identity element is a unique element in the group that, when combined with any other element using the group operation, leaves that element unchanged. It is commonly denoted by e or 1.

What is an abelian group?

An abelian group is one in which the binary operation is commutative. This means for any two elements a and b in the group, a · b = b · a

A subgroup is a subset of a group that is itself a group under the same binary operation. It must satisfy the group properties: closure, associativity, identity, and inverses.

What is a normal subgroup and how does it relate to quotient groups?

A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group. This means for every element n in the normal subgroup N and every element g in the group, gng⁻¹ is still in N. Normal subgroups allow the construction of quotient groups, where the group is partitioned into cosets of the normal subgroup.

What are group homomorphisms?

A group homomorphism is a function between two groups that preserves the group operation. This means if f: G → H is a homomorphism and a, b are elements of G, then f(a · b) = f(a) · f(b) in H.

What is Lagrange’s theorem in group theory?

Lagrange's theorem states that for any finite group, the order (number of elements) of every subgroup divides the order of the entire group. This theorem is a fundamental result in the study of finite groups.

What is Cayley’s theorem?

Cayley’s theorem states that every group is isomorphic to a subgroup of a symmetric group. This implies that every group can be represented as a group of permutations acting on a set.

How is group theory applied in other fields?

Group theory has applications in many fields including: Physics: Describing symmetries and conservation laws. Chemistry: Analyzing molecular symmetry and chemical bonding. Cryptography: Underlying structures in cryptographic systems. Mathematics: Foundational in algebra, geometry, and number theory.

Home > Docs > Group Theory > Automorphism > Automorphism Isomorphism In Non Isomorphic Groups

Automorphism Isomorphism in Non-Isomorphic Groups

Bivash Majumder

Table of Contents

Introduction

Welcome to an insightful journey into the heart of group theory, where we unravel the surprising phenomenon of automorphism isomorphism. In this lesson, we explore the curious case where two non-isomorphic groups, \(H_1\) and \(H_2\), possess automorphism groups that are isomorphic, expressed as \[ \operatorname{Aut}(H_1) \cong \operatorname{Aut}(H_2). \] This fascinating result challenges conventional perceptions and reveals deep structural similarities hidden within distinct algebraic systems.

Immerse yourself in this captivating topic as you discover detailed proofs, illustrative examples, and the elegant interplay between group structure and automorphisms. Seize the opportunity to expand your mathematical horizons and enhance your understanding of abstract algebra by exploring these profound connections.

Theorem

Find two non-isomorphic groups \(H_1\) and \(H_2\) such that \(\operatorname{Aut}(H_1)\) is isomorphic to \(\operatorname{Aut}(H_2)\).

Proof

Step 1

Let us choose the groups

Let \(H_1 = \mathbb{Z}_3\) and \(H_2 = \mathbb{Z}_4\). Since \(|\mathbb{Z}_3| = 3\) and \(|\mathbb{Z}_4| = 4\), these groups are not isomorphic.

Step 2

Let us compute their automorphism groups

For any cyclic group \(\mathbb{Z}_n\), every automorphism is determined by its action on a generator. Hence, \[ \operatorname{Aut}(\mathbb{Z}_n, +) \cong U_n, \] where \[ U_n = \{a \in \mathbb{Z}_n \mid \gcd(a, n) = 1\} \] is the group of units modulo \(n\).

Step 3

To evaluate \(\operatorname{Aut}(H_1)\) and \(\operatorname{Aut}(H_2)\)

For \(H_1 = \mathbb{Z}_3\):

\begin{align} \operatorname{Aut}(\mathbb{Z}_3) &\cong U_3 \nonumber\\ &\cong \{1,2\} \nonumber\\ &\cong \mathbb{Z}_2\nonumber \end{align}

For \(H_2 = \mathbb{Z}_4\):

\begin{align} \operatorname{Aut}(\mathbb{Z}_4) &\cong U_4\nonumber\\ &\cong \{1,3\}\nonumber\\ &\cong \mathbb{Z}_2\nonumber\\ \end{align}

Step 4

Conclude the Isomorphism of Automorphism Groups

Since both \(\operatorname{Aut}(\mathbb{Z}_3)\) and \(\operatorname{Aut}(\mathbb{Z}_4)\) are isomorphic to \(\mathbb{Z}_2\), it follows that: \[ \operatorname{Aut}(H_1) \cong \operatorname{Aut}(H_2). \]

Summary

We have selected \(H_1 = \mathbb{Z}_3\) and \(H_2 = \mathbb{Z}_4\), two non-isomorphic groups due to their differing orders. By computing their automorphism groups using the fact that \(\operatorname{Aut}(\mathbb{Z}_n, +) \cong U_n\), we showed that \(\operatorname{Aut}(\mathbb{Z}_3) \cong U_3 \cong \mathbb{Z}_2\) and \(\operatorname{Aut}(\mathbb{Z}_4) \cong U_4 \cong \mathbb{Z}_2\). Hence, the automorphism groups are isomorphic.

Automorphism

FAQs

Group Theory

What is a group in group theory?

A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for every element.
What are the main properties of a group?
The four main properties are:
- Closure: The result of the operation on any two elements of the group is also in the group.
- Associativity: The group operation is associative.
- Identity: There exists an element that does not change other elements when used in the operation.
- Invertibility: Every element has an inverse that, when combined with the element, yields the identity.
What is the identity element in a group?

The identity element is a unique element in the group that, when combined with any other element using the group operation, leaves that element unchanged. It is commonly denoted by e or 1.
What is an abelian group?

An abelian group is one in which the binary operation is commutative. This means for any two elements a and b in the group, a · b = b · a
What is a subgroup?

A subgroup is a subset of a group that is itself a group under the same binary operation. It must satisfy the group properties: closure, associativity, identity, and inverses.
What is a normal subgroup and how does it relate to quotient groups?

A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group. This means for every element n in the normal subgroup N and every element g in the group, gng⁻¹ is still in N. Normal subgroups allow the construction of quotient groups, where the group is partitioned into cosets of the normal subgroup.
What are group homomorphisms?

A group homomorphism is a function between two groups that preserves the group operation. This means if f: G → H is a homomorphism and a, b are elements of G, then f(a · b) = f(a) · f(b) in H.
What is Lagrange’s theorem in group theory?

Lagrange's theorem states that for any finite group, the order (number of elements) of every subgroup divides the order of the entire group. This theorem is a fundamental result in the study of finite groups.
What is Cayley’s theorem?

Cayley’s theorem states that every group is isomorphic to a subgroup of a symmetric group. This implies that every group can be represented as a group of permutations acting on a set.
How is group theory applied in other fields?
Group theory has applications in many fields including:
- Physics: Describing symmetries and conservation laws.
- Chemistry: Analyzing molecular symmetry and chemical bonding.
- Cryptography: Underlying structures in cryptographic systems.
- Mathematics: Foundational in algebra, geometry, and number theory.

Knowledge Bases

Previous Year Question Papers

Vidyasagar University- Quesitions Papers

Group Theory

Automorphism Isomorphism in Non-Isomorphic Groups

Introduction

Theorem

Proof

Step 1

Step 2

Step 3

Step 4

Summary

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Automorphism

FAQs

Group Theory

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